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Abstract

In this paper, we show the existence of pullback attractors for the non-autonomous
Navier-Stokes-Voight equations by using contractive functions, which is more simple
than the weak continuous method to establish the uniformly asymptotical compactness
in H1 and H2.

Here u = u(t, x) = (u1(t, x), u2(t, x), u3(t, x)) is the velocity vector field, p is the pressure, ν > 0 is the kinematic viscosity, and the length scale α is a characterizing parameter of the elasticity of the fluid.

When α = 0, the above system reduce to the well-known 3D incompressible Navier-Stokes system:

(1.5)

(1.6)

For the well-posedness of 3D incompressible Navier-Stokes equations, in 1934, Leray
[1-3] derived the existence of weak solution by weak convergence method; Hopf [4] improved Leray's result and obtained the familiar Leray-Hopf weak solution in 1951.
Since the 3D Navier-Stokes equations lack appropriate priori estimate and the strong
nonlinear property, the existence of strong solution remains open. For the infinite-dimensional
dynamical systems, Sell [5] constructed the semiflow generated by the weak solution which lacks the global regularity
and obtained the existence of global attractor of the 3D incompressible Navier-Stokes
equations on any bounded smooth domain. Chepyzhov and Vishik [6] investigated the trajectory attractors for 3D non-autonomous incompressible Navier-Stokes
system which is based on the works of Leray and Hopf. Using the weak convergence topology
of the space H (see below for the definition), Kapustyan and Valero [7] proved the existence of a weak attractor in both autonomous and non-autonomous cases
and gave a existence result of strong attractors. Kapustyan, Kasyanov and Valero [8] considered a revised 3D incompressible Navier-Stokes equations generated by an optimal
control problem and proved the existence of pullback attractors by constructing a
dynamical multivalued process.

However, the infinite-dimensional systems for 3D incompressible Navier-Stokes equations
have not yet completely resolved, so many mathematicians pay attention to this challenging
problem. In this regard, Kalantarov and Titi [9] investigated the Navier-Stokes-Voight equations as an inviscid regularization of
the 3D incompressible Navier-Stokes equations, and further obtained the existence
of global attractors for Navier-Stokes-Voight equations. Recently, Qin, Yang and Liu
[10] showed the existence of uniform attractors by uniform condition-(C) and weak continuous
method to obtain uniformly asymptotical compactness in H1 and H2, Yue and Zhong [11] investigated the attractors for autonomous and nonautonomous 3D Navier-Stokes-Voight
equations in different methods. More details about the infinite-dimensional dynamics
systems, we can refer to [12-27].

Using the contractive functions, we have in this paper established the uniformly asymptotical
compactness of the processes {U(t, τ)}(t ≥ τ, τ ∈ R) to obtain the existence of the uniform attractor of the 3D non-autonomous NSV equations.

Main difficulties we encountered are as follows:

(1) how to obtain a contractive function,

(2) how to deduce the uniformly asymptotical compactness from a contractive function,

(3) how to obtain the convergence of contractive function.

2 Main results

Notations: Throughout this paper, we set Rτ = [τ, +∞), τ ∈ R. C stands for a generic positive constant, depending on Ω, but independent of t. Lp(Ω)(1 ≤ p ≤ +∞) is the generic Lebesgue space, Hs(Ω) is the general Sobolev space. We set , H, V, W is the closure of the set E in the topology of (L2(Ω))3, (H1(Ω))3, (H2(Ω))3 respectively. "⇀" stands for weak convergence of sequence.

Let be the hull of f0 as a symbol space:

(2.1)

for all , where denotes the closure in the topology of .

Under the assumptions of the initial data, the problem (1.1)-(1.4) has a global solution
u ∈ C([τ, +∞), V). Uf(t, τ, uτ): V → V denotes the processes generated by the global solutions and satisfies

(2.2)

(2.3)

(2.4)

Let {T(s)} be the translation semigroup on Σ, we see that the family of processes {Uf(t, τ)} (f ∈ Σ) satisfies the translation identity if

(2.5)

(2.6)

Next, we recall a simple method to derive uniformly asymptotical compactness which
can be found in [28].

Definition 2.1 Let X be a Banach space and B be a bounded subset of X, Σ be a symbol space. We call a function ϕ(·,·;·,·) defined on (X × X) × (Σ × Σ) to be a contractive function on B × B if for any sequence and any {gn} ⊂ Σ, there are subsequences and such that

(2.7)

We denote the set of all contractive functions on B × B by Contr(B, Σ).

Lemma 2.2 Let {Uf(t, τ)}(f ∈ Σ) be a family of processes satisfying the translation identity (2.5) and (2.6) on Banach
space X and has a bounded uniform (w.r.t f ∈ Σ) absorbing set B0 ⊂ X. Moreover, assume that for any ε > 0, there exist T = T(B0, ε) and ϕT ∈ Contr(B0, Σ) such that

3 Proof of Theorem 2.3

In this section, we shall prove Theorem 2.3 by two steps as follows, the first one
is to get the existence of an absorbing ball, the second is to prove the asymptotical
compactness by means of a contractive function.

From the property of solutions, we can easily derive that the set class {Uf(t, τ, uτ)} (τ, ≤ t) is a process in V for all τ ≤ t. Moreover, the mapping Uf(t, τ, uτ): V → V is continuous.

Lemma 3.1 We assume that and , fn → f in L2(R, H), then

(3.1)

(3.2)

Proof. From the boundedness of the solutions in corresponding topological spaces, we easily
conclude the results. □

Hence for the above T. By Lemma 2.2 and the property of the functional ⟨·,·⟩ + α2 ⟨∇·, ∇·⟩, the conclusion holds. □

Proof of Theorem 2.3 From Lemmas 3.1-3.3, we can deduce the result easily. □

4 Proof of Theorem 2.4

Similarly to the proof of Theorem 2.3, we easily obtain that the set class {Uf (t, τ, uτ)} (τ ≤ t) is a process in W for all τ ≤ t. Moreover, the mapping Uf (t, τ, uτ): W → W is continuous. If we assume that is a sequence in W and weakly converges to uτ ∈ W, fn → f in L2(R, H), then

Proof. By the Faedo-Galerkin method, the standard elliptic operator theory and the Poincaré
inequality, we get that u belongs to L2((τ, T); D(A)) ∩ L∞((τ, T); W), then using the Gronwall inequality and similar energy method to the proof of Theorem
3.1 in Qin, Yang and Liu [10], we can deduce the boundedness of u and the existence of absorbing set. □

Proof. For any initial data , let ui(t, x) be the corresponding solutions to the symbols fi with , that is, ui(t) is the solution of the problem (3.8)-(3.11). Denote A = -Δ and w(t) = u1(t) - u2(t), then w(t) satisfies the equivalent abstract equations (3.13)-(3.14).

Setting

(4.3)

Multiplying (3.13) by Aw and integrating over [s, T] × Ω, we deduce

(4.4)

where τ ≤ s ≤ T. Then we have

(4.5)

Hence,

(4.6)

Integrating (4.4) over [τ, T] with respect to s, we get

(4.7)

If we set

(4.8)

then we have

(4.9)

Since the family of the processes has a uniformly bounded absorbing set, we choose
T large enough such that